Integral variable substitution for removing singularity

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Homework Help Overview

The original poster discusses an integral of the form \(\int\frac{A+Bx+Cx^2}{Dx+Ex^2}\), where A-E are constants, and seeks to remove a singularity at \(x=0\) through substitution. The integration limits are from 0 to 1.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts various substitutions, including squares, roots, and logarithmic functions, but encounters persistent singularities. They express uncertainty about how to effectively remove the singularity.
  • Some participants suggest specific substitutions, such as \(u = 1/x\) and polynomial long division, while noting that these may not fully resolve the singularity issue.
  • One participant mentions that the integral is under a square root, which prompts a reconsideration of the substitution approach.

Discussion Status

Participants are actively exploring different substitution methods and discussing their effectiveness in addressing the singularity. There is acknowledgment of the complexity introduced by the square root in the integral, and some suggestions have led to new insights for the original poster.

Contextual Notes

The original poster indicates that the constants A-E are lengthy and complicated, but they believe these values may not be necessary for the discussion. There is also a mention of homework constraints that limit the ability to manipulate the integral through certain algebraic methods.

Hokey
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Homework Statement
Hi! In an assignment I have reached an integral that has the form:
[itex]\int\frac{A+Bx+Cx^2}{Dx+Ex^2}[/itex]
where A-E are constants, the integration variable is x and the limits are 0 to 1. I'm supposed to remove the singularity at x=0 by substitution.

A-E have values but they're long and complicated and I hope they're not necessary to solve the problem. And sadly, no - I can't go backwards to complete squares or anything...


The attempt at a solution
This might be an easy question, but I really don't know what to substitute x for. I've tried squares, roots, inverted squares and roots, ln and exponential functions... but they all end up with the same singularity at 0. I don't know how to get around this. Is there any good way to figure out how to substitute in order to "remove" a singularity, in general?

Help and hints would be very much appreciated! Thank you!

/Jennifer
 
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Try the following. Not sure how well it works for the whole problem.

Let u = 1/x . Of course that's equivalent to x = eu .
 
Suggestion: Divide using long, polynomial division and rewrite the integral :)
 
MednataMiza said:
Suggestion: Divide using long, polynomial division and rewrite the integral :)
That still leaves a singularity at x=0.
 
Ahh. Thanks for your suggestions, they made me re-track my steps. And yep, I left out the part that the whole fraction is under a square root as well :-p That changes things a bit!

Substituting x with u2 seems to do the trick now! Phew :redface:
 

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